Of the 1,061 prime numbers between 1,000 and 10,000, how many of them are separated by EXACTLY 1,000? In other words, the difference between any 2 of them is exactly 1,000 such as: 9941 and 8941? Any help would be appreciated. Thank you.
+6251 There's no number theoretic solution to this that I know of.
You just have to go through the list and check each one.
By hand this is absurdly tedious. Something you'd give a team of 12th century monks a month to complete.
In software it's pretty trivial to code up a quick program to dump the list of them and it's length.
How many monks do you have available?
Well, as Rom says, you could list all of the 1061 prime numbers in descending order and manually check each number, one by one, and list them that way. However, it may take you quite some time to do that. Of course, the easiest way is to write a short computer code to do it for you almost instantaneously. That is just what I did in the following short code. Each pair, starting at the top, is separated by exactly 1000:
a=(9973, 9967, 9949, 9941, 9931, 9929, 9923, 9907.....etc); i=0; j=0; p=0; c=a[i] - a[j];if(c==1000, goto loop,goto next);loop:printa[i]," ",;printa[j]," ",;p=p+count(c);next:j++;if(j >Total = ",p
9941 8941 9929 8929 9923 8923 9887 8887 9839 8839 9803 8803 9719 8719 9689 8689 9677 8677 9629 8629 9623 8623 9539 8539 9521 8521 9467 8467 9461 8461 9431 8431 9419 8419 9377 8377 9311 8311 9293 8293 9221 8221 9209 8209 9161 8161 9059 8059 9011 8011 8963 7963 8951 7951 8933 7933 8867 7867 8753 7753 8741 7741 8699 7699 8681 7681 8669 7669 8573 7573 8537 7537 8369 7369 8297 7297 8243 7243 8237 7237 8219 7219 8069 7069 8039 7039 7949 6949 7907 6907 7883 6883 7841 6841 7829 6829 7823 6823 7793 6793 7703 6703 7691 6691 7673 6673 7607 6607 7577 6577 7547 6547 7529 6529 7481 6481 7451 6451 7247 6247 7229 6229 7211 6211 7151 6151 7121 6121 7079 6079 7043 6043 6869 5869 6857 5857 6827 5827 6791 5791 6779 5779 6737 5737 6701 5701 6689 5689 6659 5659 6653 5653 6581 5581 6569 5569 6563 5563 6521 5521 6449 5449 6323 5323 6197 5197 6113 5113 6101 5101 6011 5011 5987 4987 5903 4903 5861 4861 5813 4813 5801 4801 5783 4783 5657 4657 5651 4651 5639 4639 5591 4591 5519 4519 5507 4507 5483 4483 5441 4441 5297 4297 5273 4273 5261 4261 5231 4231 5153 4153 5099 4099 5051 4051 5021 4021 5003 4003 4967 3967 4943 3943 4931 3931 4919 3919 4889 3889 4877 3877 4793 3793 4733 3733 4691 3691 4673 3673 4643 3643 4637 3637 4583 3583 4547 3547 4517 3517 4463 3463 4457 3457 4391 3391 4373 3373 4271 3271 4259 3259 4253 3253 4229 3229 4217 3217 4079 3079 4049 3049 4019 3019 4001 3001 3917 2917 3851 2851 3833 2833 3803 2803 3797 2797 3767 2767 3719 2719 3677 2677 3671 2671 3659 2659 3617 2617 3593 2593 3557 2557 3539 2539 3467 2467 3389 2389 3371 2371 3347 2347 3251 2251 3221 2221 3203 2203 3137 2137 3089 2089 3083 2083 3011 2011 2999 1999 2879 1879 2861 1861 2801 1801 2789 1789 2777 1777 2753 1753 2741 1741 2699 1699 2693 1693 2663 1663 2657 1657 2621 1621 2609 1609 2579 1579 2549 1549 2543 1543 2531 1531 2459 1459 2447 1447 2423 1423 2399 1399 2381 1381 2297 1297 2237 1237 2213 1213 2153 1153 2129 1129 2087 1087 2069 1069 2063 1063 2039 1039 >>Total = 194 such numbers.
